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Questions tagged [prime-numbers]

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

A prime number (or a prime) is an element of the greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number ... The fundamental theorem of arithmetic establishes the central role of primes in :

Any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

Here you get the first 50 millions of primes.

The concept of prime numbers is extended in ring theory, where an element $p$ of a ring $R$ is prime if and only if whenever $p\mid ab$, then $p\mid a$ or $p\mid b$.

One can easily see that this extends the definition of prime numbers in the natural numbers.

12562 questions
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Largest known integer $m$ such that Legendre's conjecture is true up to $m$

The Legendre's conjecture states that the inteval $(m^2,(m+1)^2)$ contains a prime $p_n$ for each $m$. I wante to find the largest known integer $m$ such that this conjecture is true up to $m$. Wikipedia…
DER
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proof of $3k+1$ has infinite many primes by directly using the proof of there exist infinite primes

Explain why you cannot directly adapt the proof that there are infinitely many primes to show that there are infinitely many primes in the arithmetic progression 3k+1,k=1,2,... In my opinion ,I feel that the proof that there are infinitely many…
David Zhu
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How can I find the solutions of $x=2\pi(2 x)-2$, where $\pi$ is the prime counting function?

If $x>2$, $x$ is even number, and $\pi(x)$ is a number of prime numbers less than $x$, then how can I find the roots of the equation $$x=2\pi(2x)-2$$ without the use of computer programming.
Jamal Farokhi
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Prove or disprove: If $n$ is not prime, then there exists an integer $a$ which divides $n$ and is in the interval $2\leq a \leq \sqrt{n}$.

If $n$ is not prime, then there exists an integer $a$ which divides $n$ and is in the interval $2\leq a \leq \sqrt{n}$. I need to either prove or disprove this. My approach would to first attempt to test values for $a$ and $n$ to see if it seems…
klorzan
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A question based on prime numbers.

Show that if $P$ and $8P - 1$ are prime, then $8P + 1$ is composite. First of all I analyzed that except for the case of 2 & 3, the minimum difference between two prime numbers is always greater than or equal to 2. Then via shrewd deduction I found…
user247855
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the product of two prime numbers is 1994, what is their sum?

The product of two prime numbers is $1994$, what is their sum. So if you look at the factors of $1994$, you have: $1,2,997,1994$. Since $1$ is not considered prime. then the two numbers are $2$ and $997$, so their sum is $999$. But is there a way to…
Caddy Heron
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Unique partial sum of a finite prime number power series

Are there any quick and simple ways to prove or disprove that $$ \sum_{\substack{k\in G_1\\ G_1 \subset\mathbb{N}_n}} p^k \neq \sum_{\substack{j\in G_2 \\ G_2 \subset(\mathbb{N}_n\setminus G_1)}} p^j $$ where $p$ is a prime number and…
Gong-Yi Liao
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If $p_x ^2 = 2n+1$ then $n$ is always even?

If an odd prime squared $p_x ^2$ is written in the form $2n+1$, is $n$ always even? Furthermore, is it true that $k\cdot p_x + n$ is always prime when $k\cdot p_x + n < p_{x+1} ^2$ and $\gcd(k,n)=1$ where $k \in \mathbb{N}$ and $n$ is as…
Brad Graham
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Primes that are approximately twice other primes

Are there infinitely many pairs of primes of the form $p,2p-1$? What about $p,2p+1$?
msinghal
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Is there a method to determine a prime number containing the first n digits?

For example, the number $10243$ is prime and contains the digits '0,' '1,' '2,' '3,' and '4.' Similarly, the number $20143$ is prime. Is there a method to determine whether a prime number exists that contains the first, say, $8$ digits? Or…
Mary
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Finding all the divisors of $a$ by decomposing it into the product $p^{\alpha_1}_{1} \cdot p^{\alpha_2}_{2} \cdots p^{\alpha_r}_{r}$

I'm trying to prove the following statement regarding the fundamental facts of prime numbers, but I don't really understand the relationship between $a$ and $b$. In order to find all the divisors of any number $a$ we need only decompose $a$ into…
hohner
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What is the importance of prime numbers in cryptographic algorithms?

such a RSA algorithm.
simplyblue
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Prime divisors in Andy Loo's proof...

http://arxiv.org/pdf/1110.2377v1.pdf I have one more question related to that proof. Look at the definition of the symbol ${s \brace r}$ (page 4). Why if $\frac{3n}{4}
virnoy
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A few questions about Andy Loo's proof of existence of primes between 3n and 4n...

I have a few questions about Andy Loo's proof (get it here): why, for example, if $2n
virnoy
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Prime from mirror concatenation of first primes

The mirror concatenation of the first 1, 6 and 8 prime numbers with no primes being reversed is a prime ! i.e. 131175323571113 and 19171311753235711131719 are prime numbers! (beautiful primes!). After these I couldn't find other primes with such…
Gill Vania
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